This thesis describes a general method for automatic reconstruction of accurate, concise, piecewise smooth surfaces from unorganized 3D points. Instances of surface reconstruction arise in numerous scientific and engineering applications, including reverse-engineering--the automatic generation of CAD models from physical objects.
Previous surface reconstruction methods have typically required additional knowledge, such as structure in the data, known surface genus, or orientation information. In contrast, the method outlined in this thesis requires only the 3D coordinates of the data points. From the data, the method is able to automatically infer the topological type of the surface, its geometry, and the presence and location of features such as boundaries, creases, and corners.
The reconstruction method has three major phases: (1) initial surface estimation, (2) mesh optimization, and (3) piecewise smooth surface optimization. A key ingredient in phase 3, and another principal contribution of this thesis, is the introduction of a new class of piecewise smooth representations based on subdivision. The effectiveness of the three-phase reconstruction method is demonstrated on a number of examples using both simulated and real data.
Phases 2 and 3 of the surface reconstruction method can also be used to approximate existing surface models. By casting surface approximation as a global optimization problem with an energy function that directly measures deviation of the approximation from the original surface, models are obtained that exhibit excellent accuracy to conciseness trade-offs. Examples of piecewise linear and piecewise smooth approximations are generated for various surfaces, including meshes, NURBS surfaces, CSG models, and implicit surfaces.

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Surface reconstruction from unorganized points

Introduction to linear and nonlinear programming

These are the short notes for a two hour tutorial on principles and practice of computer graphics and scientific visualization. They are intended to summarize the contents of the tutorial transparencies and slides but they cannot completely replace them since restrictions in space and print quality do not permit the inclusion of figures and example images. For further reference the following standard text should be consulted: [3, 8, 5, 1, 6, 2, 9]

Computer graphics—principles and practice

A~traet--We present a powerful, enhanced multiquadrics (MQ) scheme developed for spatial approximations. MQ is a true scattered data, grid free scheme for representing surfaces and bodies in an arbitrary number of dimensions. It is continuously differentiable and integrable and is capable of representing functions with steep gradients with very high accuracy. Monotonicity and convexity are observed properties as a result of such high accuracy. Numerical results show that our modified MQ scheme is an excellent method not only for very accurate interpolation, but also for partial derivative estimates. MQ is applied to a higher order arbitrary Lagrangian-Eulerian (ALE) rezoning. In the second paper of this series, MQ is applied to parabolic, hyperbolic and elliptic partial differential equations. The parabolic problem uses an implicit time-marching scheme whereas the hyperbolic problem uses an explicit time-marching scheme. We show that MQ is also exceptionally accurate and efficient. The theory of Madych and Nelson shows that the MQ interpolant belongs to a space of functions which minimizes a semi-norm and gives credence to our results. 1. BACKGROUND The study of arbitrarily shaped curves, surfaces and bodies having arbitrary data orderings has immediate application to computational fluid-dynamics. The governing equations not only include source terms but gradients, divergences and Laplacians. In addition, many physical processes occur over a wide range of length scales. To obtain quantitatively accurate approximations of the physics, quantitatively accurate estimates of the spatial variations of such variables are required. In two and three dimensions, the range of such quantitatively accurate problems possible on current multiprocessing super computers using standard finite difference or finite element codes is limited. The question is whether there exist alternative techniques or combinations of techniques which can broaden the scope of problems to be solved by permitting steep gradients to be modelled using fewer data points. Toward that goal, our study consists of two parts. The first part will investigate a new numerical technique of curve, surface and body approximations of exceptional accuracy over an arbitrary data arrangement. The second part of this study will use such techniques to improve parabolic, hyperbolic or elliptic partial differential equations. We will demonstrate that the study of function approximations has a definite advantage to computational methods for partial differential equations. One very important use of computers is the simulation of multidimensional spatial processes. In this paper, we assumed that some finite physical quantity, F, is piecewise continuous in some finite domain. In many applications, F is known only at a finite number of locations, {xk: k = 1, 2 ..... N} where xk = x~ for a univariate problem, and Xk = (x~,yk .... )X for the multivariate problem. From a finite amount of information regarding F, we seek the best approximation which can not only supply accurate estimates of F at arbitrary locations on the domain, but will also provide accurate estimates of the partial derivatives and definite integrals of F anywhere on the domain. The domain of F will consist of points, {xk }, of arbitrary ordering and sub-clustering. A rectangular grid is a very special case of a data ordering. Let us assume that an interpolation function, f, approximates F in the sense that

Approximation scheme with applications to computational fluid-dynamics-- i surface approximations and partial derivative estimates

We present a powerful, enhanced multiquadrics (MQ) scheme developed for spatial approximations. MQ is a true scattered data, grid free scheme for representing surfaces and bodies in an arbitrary number of dimensions. It is continuously differentiable and integrable and is capable of representing functions with steep gradients with very high accuracy. Monotonicity and convexity are observed properties as a result of such high accuracy. Numerical results show that our modified MQ scheme is an excellent method not only for very accurate interpolation, but also for partial derivative estimates. MQ is applied to a higher order arbitrary Lagrangian-Eulerian (ALE) rezoning. In the second paper of this series, MQ is applied to parabolic, hyperbolic and elliptic partial differential equations. The parabolic problem uses an implicit time-marching scheme whereas the hyperbolic problem uses an explicit time-marching scheme. We show that MQ is also exceptionally accurate and efficient. The theory of Madych and Nelson shows that the MQ interpolant belongs to a space of functions which minimizes a semi-norm and gived credence to our results.

Multiquadrics—A scattered data approximation scheme with applications to computational fluid-dynamics—I surface approximations and partial derivative estimates

The multiquadric (MQ) method is an effective bivariate interpolant to three-dimensional data (xi, yi, zi), where the (xi, yi) are arbitrarily located in the plane. The accuracy of the MQ method is dependent on a user defined parameter R2, and most practitioners select R2 based upon the number of data points and the locations of the (xi, yi) in the plane. We observe that the optimal value of R2 is a strong function of the zi, and that it is essentially independent of both the number and the locationsof the (xi, yi) points. Contrary to some opinions, we observe that the MQ method can effectively interpolate to “track data”, that is, data sampled densely along tracks in the plane. Together with several other observations, we present an algorithm that generally yields an effective value for R2.

The parameter R2 in multiquadric interpolation

Techniques for the visualization of a scalar-valued function defined over a spherical domain are discussed. Functions of this type arise, for example, in the fitting of measured data on the surface of the earth. Methods for contouring and for displaying the graph of a function are presented, along with several examples. The projected graph technique can also be considered as a method for modeling closed surfaces. >

Visualizing functions over a sphere

Visualizing two types of scattered data-volumetric data sampled in a three-dimensional volume and surface-on-surface data sampled on a three-dimensional surface-is addressed. Since measurements are often sampled at discrete scattered locations, mathematical models that are defined over the entire domain and that interpolate or approximate the given scattered data are used. The modeling function can be evaluated over a grid, so that a conventional, off-the-shelf visualization tool that applies to data on a uniform grid can be used. Volumes, gradient, centroids, and other quantities can be computed from the model. Even though the two kinds of data and their problems look similar in their mathematical descriptions, they are solved by different methods, and the various solutions are visualized with methods that are quite different. >

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Visualizing and modeling scattered multivariate data

We present a new method for the selection of knot sequences for parametric spline curves. The method takes into consideration the geometry of the control points and produces quality results for a wide variety of curve fitting problems. In addition, this method is invariant with respect to affine transformations of the control points.

Knot selection for parametric spline interpolation

Various methods have been developed to control the shape of an interpolating curve for computer-aided design applications. Some methods are better suited for controlling the tension of the curve on an interval, while others are better suited for controlling the tension at the individual interpolation points. The weighted v-spline is a C1 piecewise cubic polynomial interpolant that generalizes C2 cubic splines, weighted splines, and v-splines. Shape controls are available to “tighten” the weighted v-spline on intervals and/or at the interpolation points. The mathematical theory is presented together with short algorithms for parametric interpolation.

Thomas A. Foley

Papers

The parameter R2 in multiquadric interpolation

Visualizing functions over a sphere

Visualizing and modeling scattered multivariate data

Knot selection for parametric spline interpolation

Interpolation with interval and point tension controls using cubic weighted v-splines